The present invention relates generally to servo control, and more specifically to switching processes for sensor-free servo control.
The concept of feedback control has witnessed countless advances and breakthroughs, one after another, not only in theory but in diverse applications far beyond its original engineering horizon. Its richness also has necessitated the creation of new disciplines and finer division of old disciplines, such as adaptive control, optimal control, H∞-control, and the like. Amongst the list of challenges and accomplishments from its inception, servomechanisms are one of the most fundamental problems in control theory.
In the simplest setting, a servo can be cast into a standard linear, second order, constant-coefficient ordinary differential equation most often used in modeling a mass-damper-spring system with an external excitation, also referred to in certain cases as an input or a stimulus.
Servomechanisms, or servos, are typically controlled using feedback. However, the use of feedback implies use of sensors to measure observables. Sensors have an associated cost. Sensors take up space which could otherwise be used for payload, and sensors have an associated weight that must be considered for overall performance, specifications, guidance, range, and the like. Further, sensors themselves bring uncertainty and noise into systems. Sensors placed on the servo, or incorporated into the servo, gather and relay information, typically including position, velocity, and acceleration information, to a controller through feedback paths. Such feedback leads to good control of servo operation.
Consider a standard mass-damper-spring system that is severely underdamped. Such systems suffer not only large over- and under-shoots but long settling times. The former poses danger in saturation and excites nonlinear dynamics either unconsidered or ignored, whereas the latter can induce catastrophic resonance among all such systems.
Overdamped and the critically damped systems have zero overshoot but suffer long rising time and so have even longer settling time. A related feature of these systems is their monotony. The xe2x80x98steadyxe2x80x99 state is reached only asymptotically.
There are a collection of feedback control problems which historically have been solved in this manner: (a) solving the open-loop control as a function of time U=u(t), (b) equating u as an unknown function, or operator F (linear, nonlinear) of the state x, the output y, or the measurement z, and solving for F.
Consider a basic servomechanism problem to begin with. For single input single output (SISO) linear time invariant (LTI) systems in form of
anx(n)+anxe2x88x921x(nxe2x88x921)+ . . . +a1{dot over (x)}+a0x=u(t),xe2x80x83xe2x80x83(2.1)
with a given desired system response x(t) satisfying certain essential Laplace transformability conditions, the pertinent servo control input û(t) can be shown as given by
û=Lxe2x88x921{Rn{circumflex over (X)}xe2x88x92Rnxe2x88x921x0xe2x88x92Rnxe2x88x922{dot over (x)}0xe2x88x92 . . . xe2x88x92R1x0(nxe2x88x922)xe2x88x92R0x0(nxe2x88x921)}xe2x80x83xe2x80x83(2.2)
where {circumflex over (X)} (s)=L{{circumflex over (x)}(t)}, {circumflex over (x)}(t) the desired system response, and
Rk(s):=ansk+anxe2x88x921skxe2x88x921+ . . . +anxe2x88x92kxe2x88x921s+anxe2x88x92k=sRnxe2x88x921+anxe2x88x92k, R0=an. xe2x80x83xe2x80x83(2.3)
In particular, if x0={dot over (x)}0= . . . {dot over (x)}0(nxe2x88x921)=0, then
û=Lxe2x88x921{(ansn+ . . . +a1s+a0){circumflex over (X)}}. xe2x80x83xe2x80x83(2.4)
For multiple input multiple output (MIMO) LTI systems {dot over (x)}=Ax+Bu, equation (2.2) changes to
û=(BTB)xe2x88x921BT[(sIxe2x88x92A){circumflex over (X)}xe2x88x92x0]xe2x80x83xe2x80x83(2.5)
assuming invertibility of BTB. Equations (2.2)-(2.3) are considered as a summary/solution to the most basic linear servomechanism problems. No consideration is given to conditions such as ∥u(t)∥xe2x89xa61, however.
Note that if {circumflex over (x)}(t)={circumflex over (x)}f for txe2x89xa7tf, then                                                         X              ^                        ⁡                          (              s              )                                =                                                    ∫                0                                  t                  f                                            ⁢                                                x                  ^                                ⁢                                  xe2x80x83                                ⁢                                  ⅇ                                      -                    st                                                  ⁢                                  xe2x80x83                                ⁢                                  ⅆ                  t                                                      =                                          -                                                                            ⅇ                                              -                                                  st                          f                                                                                      s                                    ⁡                                      [                                                                                            x                          ^                                                f                                            +                                                                        ∫                          0                                                      t                            f                                                                          ⁢                                                                                                                                                                                                                  x                                    ^                                                                    .                                                                ⁢                                                                  xe2x80x83                                                                ⁢                                                                  ⅇ                                                                      -                                    st                                                                                                                              ⁢                                                              xe2x80x83                                                                                                                    xe2x80x83                                                                                ⁢                                                      ⅆ                            t                                                                                                                ]                                                              =                              …                =                                  -                                                                                    ⅇ                                                  -                                                      st                            f                                                                                              s                                        ⁡                                          [                                                                                                    x                            ^                                                    f                                                +                                                                              1                            s                                                    ⁢                                                                                                                    x                                ^                                                            .                                                        f                                                                          +                                                                              1                                                          s                              2                                                                                ⁢                                                                                                                    x                                ^                                                            ¨                                                        f                                                                          +                        …                                            ]                                                                                                          ,                            (        2.6        )            
provided that (2.1) starts from rest. On the other hand, given an input û(t)⇄Û(s),                               X          ^                =                                            U              +                                                ∑                                      k                    =                    1                                    n                                ⁢                                  xe2x80x83                                ⁢                                                      R                                          n                      -                      k                                                        ⁢                                      x                    0                                          (                                              k                        -                        1                                            )                                                                                                          R              n                                =                                    ∷                        ⁢                                                            N                  ⁡                                      (                    s                    )                                                                    P                  ⁡                                      (                    s                    )                                                              .                                                          (        2.7        )            
Smith (Smith, O. J. M., Feedback Control Systems, McGraw-Hill, N.Y., 1958) was the first who proposed Posicast control (for positive-cast) and demonstrated the idea with a very simple mechanism, a pendulum. For time-optimal control systems, Neustadt (Neustadt, L. W., xe2x80x9cTime Optimal Control Systems With Position and Integrals Limits,xe2x80x9d J. Math. Anal. and Appl., Vol. 3,406-427, 1961) included position and integral limits in the Pontryagin Maximum Principle. Wang (Wang, P. K. C., xe2x80x9cAnalytical Design of Electrohydraulic Servomechanisms with Near Time-Optimal Responses,xe2x80x9d IEEE Trans. Auto. Control, Vol. 8, No. 1, 15-27, 1963) had a dual-mode closed-loop analytical design for electrohydraulic servomechanisms. Davies (Davies, R. M., xe2x80x9cAnalytical Design for Time Optimum Transient Response of Hydraulic Servomechanisms,xe2x80x9d J. Mech. Eng. Sci., Vol. 7, No. 1, 8-14, 1965) had an analytical design for hydraulic servomechanism. Being oriented toward practical application, both Wang and Davies elaborated on modeling and operation of nonlinear electrohydraulic and hydraulic servomechanisms, respectively. Athens (Athens, M., xe2x80x9cMinimum-Fuel Control of Second-Order Systems with Real Poles,xe2x80x9d IEEE Trans. Auto. Control, Vol. 9, No. 5, 148-153, 1964) derived the switching curves for minimum-fuel control of linear second-order systems with real poles. Different from minimum-time problem in which the control is necessarily bang-bang, the minimum-fuel control results in a bang-zero-bang profile. Of related interest, Ellert and Merriam (Ellert, F. J., and Merriam, C. W., xe2x80x9cSynthesis of Feedback Controls Using Optimization Theoryxe2x80x94An Example,xe2x80x9d IEEE Trans. Auto. Control, Vol. 8, No. 4, 89-103, 1963) employed the so-called Parametric Expansion Method to vary the weighting factors for synthesis of linear time-varying feedback controls using optimization theory and illustrated by designing an aircraft landing system.
Yastreboff (Yastreboff, M., xe2x80x9cSynthesis of Time-Optimal Control by Time Interval Adjustment,xe2x80x9d IEEE Trans. Auto. Control, Vol. 14, No. 12, 707-710, 1969) seemed to have initiated synthesis of time-optimal control for LTI systems with real modes by time interval adjustment. It has appeared that Goldwyn-Sriram-Graham (Goldwyn, R. M., Sriram, K. P., and Graham, M., J. SIAM Control, Vol. 5, 295, 1967) was the first which explicitly considered the switching times as unknowns to solve. Davison and Monro (Davison, E. J., and Monro, D. M., xe2x80x9cA Computational Technique for Finding xe2x80x9cBang-Bangxe2x80x9d Controls of Non-Linear Time-Varying Systems,xe2x80x9d Automatica, Vol. 7, 255-260, 1971) gave a hill climbing-based computational technique finding the bang-bang control switching times of nonlinear time-varying systems. Farlow (Farlow, F. J., xe2x80x9cOn Finding Switching Times in optimal Control Systems,xe2x80x9d Int. J. Control, Vol. 17, No. 4, 855-861, 1973) extended Goldwyn-Sriram-Graham and used complex analysis to form and solve the switching times for LTI control systems with real poles. Consider now a unit-ON/OFF input                               u          ^                =                                            1              -                              2                ⁢                                  u                  ⁡                                      (                                          t                      -                                              t                        1                                                              )                                                              +                              2                ⁢                                  u                  ⁡                                      (                                          t                      -                                              t                        2                                                              )                                                              -              …              +                              2                ⁢                                                      (                                          -                      1                                        )                                                                              n                      s                                        -                    1                                                  ⁢                                  u                  ⁡                                      (                                          t                      -                                              t                                                                              n                            s                                                    -                          1                                                                                      )                                                              +                                                                    (                                          -                      1                                        )                                                        n                    s                                                  ⁢                                  u                  ⁡                                      (                                          t                      -                                              t                                                  n                          s                                                                                      )                                                                        ↔                          U              ^                                =                                    1              s                        ⁡                          [                              1                -                                  ⅇ                                      -                                          st                      1                                                                      +                                  2                  ⁢                                      ⅇ                                          -                                              st                        2                                                                                            -                …                +                                  2                  ⁢                                                            (                                              -                        1                                            )                                                                                      n                        s                                            -                      1                                                        ⁢                                      ⅇ                                          -                                              st                                                                              n                            s                                                    -                          1                                                                                                                    +                                                                            (                                              -                        1                                            )                                                              n                      s                                                        ⁢                                      ⅇ                                          -                                              st                                                  n                          s                                                                                                                                ]                                                          (        2.8        )            
where t1, . . . , tns=: tf are the unknowns. Farlow reasoned that they can be solved thru equations             lim              s        →                  s          k                      ⁢          N      ⁢              (        s        )              =  0
where
P(sk)=0, k=1, . . . , n,xe2x80x83xe2x80x83(2.9)
by applying a property of removable singularities for complex entire functions. Farlow restriced this method to real sk only. In addition, it does not appear that (2.9) is solvable (with 0 less than t1 less than  . . .  less than tf) for any given initial condition. Recent attention of minimum-time and/or of minimum-fuel control has been centered at flexible structures. Singh, Kabamba, and McClamroch pointed out an important time-symmetry property in planar, time-optimal, rest-to-rest stewing maneuvers of flexible spacecraft (Singh, G., Kabamba, P. T., and McClamroch, N. H., xe2x80x9cPlanar, Time-Optimal, Rest-to-Rest Slewing Maneuvers of Flexible Spacecraft,xe2x80x9d AIAA J. Guidance, Vol. 12, No. 1, 71-81, 1989). Focused on reducing or eliminating the endpoint vibration, Singer and Seering took an input-shaping approach making use of multiple impulses and convolution and showed its robustness to parameter variations (Singer, N. C., and Seering, W. P., xe2x80x9cPreshaping Command Inputs to Reduce System Vibration,xe2x80x9d Trans ASME, Vol. 112, 76-82, March, 1990). More recently, Pao led extensive studies in this area showing the equivalence between minimum-time input shaping and traditional time-optimal control [Pao-Singhose 1995], comparing constant and variable amplitude input shaping methods for vibration reduction [Pao-Singhose 1995], and obtaining minimum-time control characteristics of flexible structures (Pao, L. Y., xe2x80x9cMinimum-Time Control Characteristics of Flexible Structures,xe2x80x9d AIAA J. Guidance, Control, and Dynamics, Vol. 19, No. 1, 123-129, 1996). Earlier, Barbieri and xc3x96zgxc3xcner also proposed a new minimum-time control law for a one-mode model of a flexible slewing structure (Barbieri, E., and xc3x96zgxc3xcner, xc3x9c., xe2x80x9cA New Minimum-Time Control Law for a One-Mode Model of a Flexible Slewing Structure,xe2x80x9d IEEE Trans. Auto. Control, Vol. 38, No. 1, 12-146, 1993).
The control systems in the papers mentioned above are all in the form of
{dot over (x)}=Ax+buxe2x80x83xe2x80x83(2.10)
where
A=[A0xc3x97A1xc3x97 . . . xc3x97An] and b=[0;b0;0;b1; . . . ;0;bn],xe2x80x83xe2x80x83(2.11)
with             A      0        =          (                                    0                                1                                                0                                0                              )        ,            b      0        =    1    ,            A      i        =          (                                    0                                1                                                              -                              ω                i                2                                                                                        -                2                            ⁢                              ξ                i                            ⁢                              ω                i                                                        )        ,
i=1, . . . ,n (x and; denoting (block) diagonal and column formations, respectively). That is, n oscillatory modes are modeled in, or rather added into, consideration in addition to the main, double-integrator. There, the pivotal equation                     {                                                                                                                                                                          1                          -                                                      2                            ⁢                                                          ⅇ                                                                                                ω                                  i                                                                ⁢                                                                  s                                  i                                                                ⁢                                                                  t                                  1                                                                                                                                              +                          …                          +                                                      2                            ⁢                                                                                          (                                                                  -                                  1                                                                )                                                                                            n                                s                                                                                      ⁢                                                          ⅇ                                                                                                ω                                  i                                                                ⁢                                                                  s                                  i                                                                ⁢                                                                  t                                                                      n                                    s                                                                                                                                                                                +                                                                                                                    (                                                                  -                                  1                                                                )                                                                                                                              n                                  s                                                                +                                1                                                                                      ⁢                                                          ⅇ                                                                                                ω                                  i                                                                ⁢                                                                  s                                  i                                                                ⁢                                                                  t                                  f                                                                                                                                                                    =                        0                                            ,                                                                                                                                                                                    1                          -                                                      2                            ⁢                                                          ⅇ                                                                                                ω                                  i                                                                ⁢                                                                                                      s                                    _                                                                    i                                                                ⁢                                                                  t                                  1                                                                                                                                              +                          …                          +                                                      2                            ⁢                                                                                          (                                                                  -                                  1                                                                )                                                                                            n                                s                                                                                      ⁢                                                          ⅇ                                                                                                ω                                  i                                                                ⁢                                                                                                      s                                    _                                                                    i                                                                ⁢                                                                  t                                                                      n                                    s                                                                                                                                                                                +                                                                                                                    (                                                                  -                                  1                                                                )                                                                                                                              n                                  s                                                                +                                1                                                                                      ⁢                                                          ⅇ                                                                                                ω                                  i                                                                ⁢                                                                                                      s                                    _                                                                    i                                                                ⁢                                                                  t                                  f                                                                                                                                                                    =                        0                                            ,                                                                                  ⁢                              
                            ⁢              i                        =            1                    ,          …          ⁢                      xe2x80x83                    ,          n          ,                                    (        2.12        )            
xe2x80x83with k the number of switches,             s      i        :=                  p        i            +                                    p            i            2                    -          1                      ,
0 less than t1 less than  . . .  less than tn, the switching times and tf:=tns+1, were stated without derivation.
In addition, all the mentioned literature considered the control problem as a stabilization, viz. return to zero problem. In spite of their mathematical equivalence, a servo perspective is different from a stabilization viewpoint.
Damped Phase, Error Window, and Damped Settling Time
A standard underdamped second order linear control system is modeled as follows:
xxe2x80x3+2pxcfx89xxe2x80x2+xcfx892x=xcfx892A, x(0)=x0, xxe2x80x2(0)=y0,xe2x80x83xe2x80x83(3.1)
where p less than 1 and xcfx89 greater than 0. Let A greater than 0 without loss of generality. The solution x and its time-derivatives xxe2x80x2 and xxe2x80x3, also commonly interpreted as position, velocity, and acceleration, are given by                               x          =                      A            -                                          ⅇ                                                      -                    r                                    ⁢                                      xe2x80x83                                    ⁢                  φ                                            ⁢                              {                                                                            (                                              A                        -                                                  x                          0                                                                    )                                        ⁢                    cos                    ⁢                                          xe2x80x83                                        ⁢                    φ                                    -                                                            [                                                                                                    y                            0                                                                                q                            ⁢                                                          xe2x80x83                                                        ⁢                            w                                                                          -                                                  r                          ⁡                                                      (                                                          A                              -                                                              x                                0                                                                                      )                                                                                              ]                                        ⁢                    sin                    ⁢                                          xe2x80x83                                        ⁢                    φ                                                  }                                                    ,                            (        3.2        )                                          x          xe2x80x2                =                                            ⅇ                                                -                  r                                ⁢                                  xe2x80x83                                ⁢                φ                                      ⁢                          {                                                                    y                    0                                    ⁢                  cos                  ⁢                                      xe2x80x83                                    ⁢                  φ                                +                                                      (                                                                  r                        ⁢                                                  xe2x80x83                                                ⁢                                                  y                          0                                                                    +                                              d                        ⁢                                                  xe2x80x83                                                ⁢                                                  ω                          ⁡                                                      (                                                          A                              -                                                              x                                0                                                                                      )                                                                                                                )                                    ⁢                  sin                  ⁢                                      xe2x80x83                                    ⁢                  φ                                            }                                =                                    ∷                        ⁢                          xe2x80x83                        ⁢                          y              .                                                          (        3.3        )            
The following quantities are defined:                               [                      q            ,            r            ,            d            ,            φ            ,            τ                    ]                :=                  [                                                    1                -                                  p                  2                                                      ,                          p              q                        ,                          1              q                        ,                          q              ⁢                              xe2x80x83                            ⁢              ω              ⁢                              xe2x80x83                            ⁢              t                        ,                          π                              q                ⁢                                  xe2x80x83                                ⁢                ω                                              ]                                    (        3.4        )            
associated with the identities:                                                         p              2                        +                          q              2                                =          1                ,                  p          =                                    q              ⁢                              xe2x80x83                            ⁢              r                        =                          r                                                1                  +                                      r                    2                                                                                      ,                  q          =                      1            d                          ,                  
                ⁢                  d          =                                    1              +                              r                2                                                    ,                                            q              2                        ⁡                          (                              1                +                                  r                  2                                            )                                =          1.                                    (        3.5        )            
Note also that                               0           less than           α                :=                                            cos                              -                1                                      ⁢            p                    =                                                    sin                                  -                  1                                            ⁢              q                        =                                                            tan                                      -                    1                                                  ⁢                                  q                  p                                            =                                                                    cot                                          -                      1                                                        ⁢                  r                                 less than                                                       π                    2                                    .                                                                                        (        3.6        )            
Traditionally, consideration of p is centered about 1, i.e., classifying (3.1) into being underdamped with p less than 1, pardamped (critically damped) with p=1 and overdamped with p greater than 1. From a qualitative (stability) viewpoint, p=0, viz. r=0 is the criticality: x(t) crosses A infinitely many times; pxe2x86x921, viz. rxe2x86x92∞ is the limit (that x(t) crosses A infinitely many times); and p greater than 1 renders the number of these crossings to only zero or possibly only one or two (but never more).
It will be clear that working with the damped phase (angle) xcfx86 is more than with time t directly. In addition, xcex1 is monotone, decreasing in p, viz. the more severely underdamped (3.1) is, the closer xcex1 is to its maximum xcfx80/2. Another significance of p involves maximum velocity; see Remark 4.1.
It is common that (3.1) is considered as a servomechanism by which in minimum time possible, (3.1) settles in a specified error window about the command value A. To specify such a window and this minimum time, also known as the settling time, denote the absolute and relative errors by E=E(t):=|Axe2x88x92x| and xcex5=xcex5(t):=E(t)/|A|. The error window may be specified either by the absolute excursion {overscore (E)} or by the relative excursion {overscore (xcex5)}, that Exe2x89xa6{overscore (E)} or xcex5xe2x89xa6{overscore (xcex5)} for all {overscore (t)}xe2x89xa6t (and the settling time is the minimum of such {overscore (t)}). For instance, x is the commanded angle of attack xcex1, {overscore (E)} might be 0.1xc2x0{fraction (xcfx80/180)}, while {overscore (xcex5)} might be, say, 5% of the commanded angle of attack.
The notion of precision                               υ          _                :=                              1                          ε              _                                =                                    E              _                                      |              A              |                                                          (        3.7        )            
then follows naturally. This notion leads further to the power of precision
"sgr":=ln{overscore (xcexd)},xe2x80x83xe2x80x83(3.8)
an equivalent of the binary, index of difficulty I:=log2 {overscore (xcexd)}, also identified as bits of information. Note that I="sgr"log2e=1.442695"sgr".
With all system-intrinsic parameters considered, the settling time of (3.1), denoted by ts, is a function of {overscore (s)}, or any of its equivalents, {overscore (xcex5)}, "sgr", etc. Besides, it is self-evident that ts also depends on the initial condition [x0, {dot over (y)}0] of (3.1). To determine the settling time of (3.1) given an initial condition [x0, y0], process flows from the simplest state, also known as the zero state, with [x0, y0]=[0, 0] to the most general cases.
With [x0, y0]=[0, 0], [x, y] reduces to
[x,y]=A[1xe2x88x92exe2x88x92rxcfx86(cos xcfx86+r sin xcfx86),dxcfx89exe2x88x92rxcfx86sin xcfx86].xe2x80x83xe2x80x83(3.9)
Evidently, when y=0, x has its turning (maximum or minimum) points, at which {overscore (xcfx86)}k=kxcfx80 and
{overscore (x)}k=A[1+(xe2x88x921)k+1xcexrk], k=1,2, . . . .xe2x80x83xe2x80x83(3.10)
where
xcex:=exe2x88x92xcfx80.
It is easy, then, to see that the overshoots and the undershoots occur at k=1,3,5,7, . . . and k=2,4,6,8, . . . , respectively, with excursion magnitudes Axcexkr.
By (3.10), Aexe2x88x92krxcfx80xe2x89xa6{overscore (E)} and exe2x88x92krxcfx80xe2x89xa6{overscore (xcex5)} for [x0, y0]=[0, 0]. It follows that                                           φ            _                    =                                                    π                ⁢                                  ⌈                                      σ                                          r                      ⁢                                              xe2x80x83                                            ⁢                      π                                                        ⌉                                            ⇔                              t                _                                      =                                                            π                                      q                    ⁢                                          xe2x80x83                                        ⁢                    ω                                                  ⁢                                  ⌈                                                            ln                      ⁡                                              (                                                  1                          /                                                      ε                            _                                                                          )                                                                                    r                      ⁢                                              xe2x80x83                                            ⁢                      π                                                        ⌉                                            =                                                τ                  ⁢                                      ⌈                                          σ                                              r                        ⁢                                                  xe2x80x83                                                ⁢                        π                                                              ⌉                                                  ≥                τ                 greater than                                   π                  ω                                                                    ,                            (        3.11        )            
where ┌(xc2x7)┐ is the smallest integer greater than (xc2x7). The term ┌{fraction ("sgr"/rxcfx80)}┐ gives the total count of excursions (peaks and valleys) about A when xe2x80x98settled.xe2x80x99
If x0xe2x89xa00 and y0=0, (3.2)-(3.3) become
[x,y]=A[1xe2x88x92xcex50exe2x88x92rxcfx86(cos xcfx86r sin xcfx86), dxcfx89xcex50 exe2x88x92rxcfx86 sin xcfx86]xe2x80x83xe2x80x83(3.12)
(xcex50=(Axe2x88x92x0)/A by definition). Therefore, xcex50 if x0 greater than A, in which case the excursion sequence reverses that for xcex5 greater than 0. It is easy to see that {overscore (xcfx86)}k=kxcfx80 still while {overscore (x)}k becomes
{overscore (X)}k=A[1+(xe2x88x921)k+1xcex50xcexrk], k=1,2, . . . .xe2x80x83xe2x80x83(3.13)
It follows that                               t          _                =                                            π                              q                ⁢                                  xe2x80x83                                ⁢                ω                                      ⁢                          ⌈                                                ln                  ⁡                                      (                                                                  |                                                  ε                          0                                                |                                                                    ε                        _                                                              )                                                                    r                  ⁢                                      xe2x80x83                                    ⁢                  π                                            ⌉                                =                                    π                              q                ⁢                                  xe2x80x83                                ⁢                ω                                      ⁢                                          ⌈                                                                            σ                      +                      ln                                        |                                          ε                      0                                        |                                                        r                    ⁢                                          xe2x80x83                                        ⁢                    π                                                  ⌉                            .                                                          (        3.14        )            
The excursions in this case are |Axe2x88x92x0| xcexrk. Note that {overscore (t)}=0 if |xcex50|xe2x89xa6{overscore (xcex5)}; (3.14) still applies.
For x0=0 and y0xe2x89xa00, (3.2)-(3.3) reduce to
[x,y]=A[{1exe2x88x92r"PHgr"(cos "PHgr"+(rxe2x88x92dxcex7)sin "PHgr"), dxcfx89exe2x88x92r"PHgr"(bcos "PHgr"+asin "PHgr")}xe2x80x83xe2x80x8393.15)
where
                              [                      η            ,            a            ,            b            ,            c            ,                          cos              ⁢                              xe2x80x83                            ⁢              β                        ,                          sin              ⁢                              xe2x80x83                            ⁢              β                                ]                :=                              [                                                            y                  0                                                  ω                  ⁢                                      xe2x80x83                                    ⁢                  A                                            ,                              1                -                                  p                  ⁢                                      xe2x80x83                                    ⁢                  η                                            ,                              q                ⁢                                  xe2x80x83                                ⁢                η                            ,                                                                    a                    2                                    +                                      b                    2                                                              ,                              -                                  a                  c                                            ,                              b                c                                      ]                    .                                    (        3.16        )            
Note that bxe2x89xa00 and
1xe2x88x922pxcex7+xcex82=1xe2x88x922pxcex7+p2xcex72+q2xcex72=a2+b2=c2.xe2x80x83xe2x80x83(3.17)
Thus, the excursions occur at the positive solutions of
a sin xcfx86+b cos xcfx86=0⇄b cos xcfx86=xe2x88x92a sin xcfx86.xe2x80x83xe2x80x83(3.18)
For {overscore (x)}1 greater than A (the first overshoot), however, it requires that                                                                         cos                ⁢                                  xe2x80x83                                ⁢                                                      φ                    _                                    1                                            +                                                (                                      r                    -                                          η                      q                                                        )                                ⁢                                  xe2x80x83                                ⁢                sin                ⁢                                  xe2x80x83                                ⁢                                                      φ                    _                                    1                                                       less than             0                    ⇒                                    -                                                1                  -                                      2                    ⁢                    p                    ⁢                                          xe2x80x83                                        ⁢                    η                                    +                                      η                    2                                                  η                                      ⁢            sin            ⁢                          xe2x80x83                        ⁢                                          φ                _                            1                                      =                                                            -                                                                            c                      2                                        ⁢                    b                                    q                                            ⁢              sin              ⁢                              xe2x80x83                            ⁢                                                φ                  _                                1                                       less than             0                    ⇒                                    b              ⁢                              xe2x80x83                            ⁢              sin              ⁢                              xe2x80x83                            ⁢                                                φ                  _                                1                                       greater than             0.                                              (        3.19        )            
It follows that                     {                                                                                                  cos                    ⁢                                          xe2x80x83                                        ⁢                                                                  φ                        _                                            1                                                        =                                      -                                          a                      c                                                                      ,                                                                                                          sin                  ⁢                                      xe2x80x83                                    ⁢                                                            φ                      _                                        1                                                  =                                  +                                                            b                      c                                        .                                                                                                          (        3.20        )            
Solving for (3.20) requires some care due to the number of sign combinations of (a,b) possible.
The minimum solutions {overscore (xcexd)} greater than 0 and xcexd{haeck over ( )} greater than 0, respectively, of the systems of trigonometric equations             [                        cos          ⁢                      xe2x80x83                    ⁢          θ                ,                  sin          ⁢                      xe2x80x83                    ⁢          θ                    ]        =                            [                                    -                              d                                                                            d                      2                                        +                                          n                      2                                                                                            ,                          +                              n                                                                            d                      2                                        +                                          n                      2                                                                                                    ]                ⁢                  xe2x80x83                ⁢                  and          ⁢                      
                    [                                    cos              ⁢                              xe2x80x83                            ⁢              θ                        ,                          xe2x80x83                        ⁢                          sin              ⁢                              xe2x80x83                            ⁢              θ                                ]                    =              [                              +                          d                                                                    d                    2                                    +                                      n                    2                                                                                ,                      -                          n                                                                    d                    2                                    +                                      n                    2                                                                                      ]              ⁢      xe2x80x83    ,
are given by                               v          ^                :=                  {                                                                                                                v                      ,                                                                                                                                                    if                          ⁢                                                      xe2x80x83                                                    ⁢                                                      (                                                          d                              ,                              n                                                        )                                                                          =                                                  (                                                      -                                                          ,                              +                                                                                )                                                                    ,                                                                                                                                                          π                        -                        v                                            ,                                                                                                                                                    if                          ⁢                                                      xe2x80x83                                                    ⁢                                                      (                                                          d                              ,                              n                                                        )                                                                          =                                                  (                                                      +                                                          ,                              +                                                                                )                                                                    ,                                                                                                                                                          π                        -                        v                                            ,                                                                                                                                                    if                          ⁢                                                      xe2x80x83                                                    ⁢                                                      (                                                          d                              ,                              n                                                        )                                                                          =                                                  (                                                      +                                                          ,                              -                                                                                )                                                                    ,                                                                                                                                                                                    2                          ⁢                          π                                                +                        v                                            ,                                                                                                                          if                        ⁢                                                  xe2x80x83                                                ⁢                                                  (                                                      d                            ,                            n                                                    )                                                                    =                                                                        (                                                      -                                                          ,                              -                                                                                )                                                .                                                                                                        ⁢                              xe2x80x83                            ⁢                              
                            ⁢              and              ⁢                              "IndentingNewLine"                            ⁢                        :=                          {                                                                                                                  π                        +                        v                                            ,                                                                                                                                                    if                          ⁢                                                      xe2x80x83                                                    ⁢                                                      (                                                          d                              ,                              n                                                        )                                                                          =                                                  (                                                      -                                                          ,                              +                                                                                )                                                                    ,                                                                                                                                                                                    2                          ⁢                          π                                                -                        v                                            ,                                                                                                                                                    if                          ⁢                                                      xe2x80x83                                                    ⁢                                                      (                                                          d                              ,                              n                                                        )                                                                          =                                                  (                                                      +                                                          ,                              +                                                                                )                                                                    ,                                                                                                                                                          -                        v                                            ,                                                                                                                                                    if                          ⁢                                                      xe2x80x83                                                    ⁢                                                      (                                                          d                              ,                              n                                                        )                                                                          =                                                  (                                                      +                                                          ,                              -                                                                                )                                                                    ,                                                                                                                                                          π                        +                        v                                            ,                                                                                                                          if                        ⁢                                                  xe2x80x83                                                ⁢                                                  (                                                      d                            ,                            n                                                    )                                                                    =                                                                        (                                                      -                                                          ,                              -                                                                                )                                                .                                                                                                        ⁢                              xe2x80x83                                                                        (        3.21        )            
where                     v        :=                              sin                          -              1                                ⁢                                    n                                                                    n                    2                                    +                                      d                    2                                                                        .                                              (        3.22        )            
By Lemma 3.1 and with similar reasoning for undershoot,                                           cos            ⁢                          xe2x80x83                        ⁢                                          φ                _                            k                                =                                                    (                                  -                  1                                )                                            k                -                1                                      ⁢                          (                              -                                  a                  c                                            )                                      ,                  
                ⁢                              sin            ⁢                          xe2x80x83                        ⁢                                          φ                _                            k                                =                                                                                          (                                          -                      1                                        )                                                        k                    -                    1                                                  ⁢                                  b                  c                                            ⇒                                                φ                  _                                k                                      =                                          β                ^                            +                                                (                                      k                    -                    1                                    )                                ⁢                π                                                    ,                  
                ⁢                  k          =          1                ,        2        ,        3        ,                  …          ⁢                      xe2x80x83                    .                                    (        3.23        )            
Note that as y0xe2x86x920, axe2x86x921 and bxe2x86x920, so xcex2xe2x86x920 while {circumflex over (xcex2)}xe2x86x92xcfx80. Thus,
{overscore (x)}k=A[1+(xe2x88x921)kxe2x88x921cexe2x88x92r{circumflex over (xcex2)}xcexrkxe2x88x92r], k=1,2,3, . . . xe2x80x83xe2x80x83(3.24)
with excursion magnitudes Acexe2x88x92r{circumflex over (xcex2)}xcexrkxe2x88x92r, which lead to                               t          _                =                                            1                              q                ⁢                                  xe2x80x83                                ⁢                ω                                      ⁢                          xe2x80x83                        ⁢                          {                              (                                                      β                    ^                                    +                                      π                    ·                                          ⌈                                                                                                    ln                            ⁡                                                          (                                                              1                                /                                                                  ε                                  _                                                                                            )                                                                                +                                                      ln                            ⁢                                                          xe2x80x83                                                        ⁢                            c                                                    -                                                      r                            ⁢                                                          xe2x80x83                                                        ⁢                                                          β                              ^                                                                                                                                r                          ⁢                                                      xe2x80x83                                                    ⁢                          π                                                                    ⌉                                                                                  }                                =                                    1                              q                ⁢                                  xe2x80x83                                ⁢                ω                                      ⁢                                          {                                                      β                    ^                                    +                                      π                    ·                                          ⌈                                                                        σ                          +                                                      ln                            ⁢                                                          xe2x80x83                                                        ⁢                            c                                                    -                                                      r                            ⁢                                                          xe2x80x83                                                        ⁢                                                          β                              ^                                                                                                                                r                          ⁢                                                      xe2x80x83                                                    ⁢                          π                                                                    ⌉                                                                      }                            .                                                          (        3.25        )            
Note that a and b cannot be both negative. Another important phase angle xcex2 also has emerged. Inevitably, however, the multiple sign combinations of (a,b) render analysis and computation for the x turning points complicated.
As an example, the case {circumflex over (xcex2)}=xcfx80xe2x88x92xcex2 can be obtained as follows. For a greater than 0 and b less than 0, xcex7 less than 0. So x(t) actually decreases before it turns toward A; that is, the first x-turning point, which occurs at xcfx86=xcex2, has |Axe2x88x92x| greater than A and is not pertinent to settling time consideration. The next xe2x80x98chancexe2x80x99 of x-turning takes place xcfx80 later.

For x0xe2x89xa00 and y0xe2x89xa00, (3.2)-(3.3) become
[x,y]=A[1xe2x88x92exe2x88x92rxcfx86(xcex50 cos xcfx86+(rxcex50xe2x88x92dxcex7) sin xcfx86), dxcfx89exe2x88x92rxcfx86(b cos xcfx86+xe2x80x2a sin xcfx86)]xe2x80x83xe2x80x83(3.26)
where a and c are adapted to                               [                                    a              xe2x80x2                        ,                          c              xe2x80x2                        ,                          cos              ⁢                              xe2x80x83                            ⁢                              β                xe2x80x2                                      ,                          sin              ⁢                              xe2x80x83                            ⁢                              β                xe2x80x2                                              ]                :=                              [                                          a                -                                                      x                    0                                    A                                            ,                                                                    a                    xe2x80x22                                    +                                      b                    2                                                              ,                              -                                                      a                    xe2x80x2                                                        c                    xe2x80x2                                                              ,                              b                                  c                  xe2x80x2                                                      ]                    .                                    (        3.27        )            
The x-turning point occurs at {overscore (xcfx86)}k={circumflex over (xcex2)}xe2x80x2+(kxe2x88x921)xcfx80 and
{overscore (x)}k=A[1+(xe2x88x921)kxe2x88x921cxe2x80x2exe2x88x92r({circumflex over (xcex2)} greater than +(kxe2x88x921)xcfx80)], k=1,2,3, . . . ,xe2x80x83xe2x80x83(3.28)
with excursion magnitudes Acxe2x80x2exe2x88x92r{circumflex over (xcex2)}xe2x80x2xcexrkxe2x88x92r, and it follows that                               t          _                =                                            π                              q                ⁢                                  xe2x80x83                                ⁢                ω                                      ⁢                          (                                                                    β                    xe2x80x2                                    ^                                +                                  ⌈                                                                                    ln                        ⁡                                                  (                                                      1                                                          ε                              _                                                                                )                                                                    +                                              ln                        ⁢                                                  xe2x80x83                                                ⁢                                                  c                          xe2x80x2                                                                    -                                              r                        ⁢                                                  xe2x80x83                                                ⁢                                                                              β                            ^                                                    xe2x80x2                                                                                                            r                      ⁢                                              xe2x80x83                                            ⁢                      π                                                        ⌉                                            )                                =                      τ            ⁢                          xe2x80x83                        ⁢                                          (                                                                            β                      xe2x80x2                                        ^                                    +                                      ⌈                                                                  σ                        +                                                  ln                          ⁢                                                      xe2x80x83                                                    ⁢                                                      c                            xe2x80x2                                                                          -                                                  r                          ⁢                                                      xe2x80x83                                                    ⁢                                                                                    β                              ^                                                        xe2x80x2                                                                                                                      r                        ⁢                                                  xe2x80x83                                                ⁢                        π                                                              ⌉                                                  )                            .                                                          (        3.29        )            
Note that the additional complexity with xcex50xe2x89xa01 has allowed (axe2x80x2,b) be (xe2x88x92,xe2x88x92).
Theorem 3.1. Consider the settling time of (3.1). Then: {overscore (t)}xe2x89xa7xcfx84 greater than {fraction (xcfx80/xcfx89)} for any [x0,y0] and
(a) if x0=0 and y0=0, the excursions are xcexrk|A| and {overscore (t)} is given by (3.11).
(b) if x0xe2x89xa00 but y0=0, the excursions are xcexrk|Axe2x88x92x0| and {overscore (t)} is given by (3.15).
(c) if x0=0 but y0xe2x89xa00, the excursions are xcexrkxe2x88x92r|A|cexe2x88x92r{circumflex over (xcex2)} and {overscore (t)} is given by (3.25).
(d) if x0xe2x89xa00 and y0xe2x89xa00, xcexrkxe2x88x92r|A|cxe2x80x2exe2x88x92r{circumflex over (xcex2)} and {overscore (t)} is given by (3.29).
As an example, say p={square root over (3)}/2 (slightly underdamped), hence q=xc2xd, ts will be at least the full natural period. Long settling time is inevitable if (3.1) is not xe2x80x98attended,xe2x80x99 that is, if some certain admissible measure, such as switching, is not applied. Above, {overscore (xcex5)}=exe2x88x923≅5% (hence, "sgr"=3 and I=4.322), p=0.1 (hence, q=0.9950 and r=0.1005) and xcfx89=1 whenever numerical comparison is called for. Thus, ts=31.5742, and it would take up to 30 ups-and-downs to settle in.
Therefore, there is a need in the art for methods and apparatuses for reducing settling time in servos, without using costly and space consuming sensors.
In one embodiment, a method for controlling a servo includes calculating at least a first switching time to change an amplitude of a command signal to a servo, providing a first amplitude to the servo for the first switching time, and switching the amplitude of the command signal to the servo from the first amplitude to a second amplitude at the first switching time.
In another embodiment, a servo system includes a servo, a processor operatively connected to control the servo, and a timer programmed to switch an input command signal amplitude at predetermined switching times.
In yet another embodiment, a command program for switching a servo without sensors includes machine readable instructions for effecting a method which includes calculating at least a first switching time to switch a command amplitude from a first level to a second level, and switching the amplitude of the command signal to the servo at the switching time.
In still another embodiment, a method of switching a servo includes applying predetermined input values to the servo at predetermined times, and settling the servo using only the input values and switching times.
Other embodiments are described and claimed.